MATH 132 EXAM II Solutions FALL 2010 1 Evaluate 9 B B E 9 B 9 B G F 9 B 9 B G G B 9 B G H 38 B 38 B G I 38 B 38 B G J B 38 B G K 9 B G L 9 B 9 B 38 B G M 38 B 38 B 9 B G N 38 B 9 B G solution 9 B 9 B B 38 B 9 B B 38 B 9 B B 38 B 38 B G I 2 What is the form of the integral substitution B 38 B B B after we make the trigonometric E 38 F 9 G 38 H 9 I 38 9 J 38 9 K 38 9 L 8 M J 8 solution 38 38 9 38 G G Use partial fractions to evaluate B B B B B E 68 B F 68 B B G 8 B H 8 B I B B B J 68 B B K 68 B L 68 B M 68 8 B N 8 6 8 B B solution B B B 68 B 68 B 68 B G K 4 Approximate B B using the Trapezoidal Rule with 8 to 4 decimal places E 5 F G H I J B K L M N B solution M 2 5 How large should n be to guarantee that Simpson s Rule of approximation for B B is acurate to within 0 0001 recall EW l O 8 E F G H I J K L M N solution 0 B B 98 O We want 8 X 2 8 8 8 L 8 6 Evaluate the improper integral B B B if it is convergent If not say it is divergent E F G H I J K L M N 3 1 8 solution B B B B l lim E 7 Find the area of the region enclosed by the curves C B 8 C B B E F G H I J K L M N solution Intersection points are 0 0 and 2 4 0 B B B is upper curve and 1 B B is the lower curve Then Area B B B B B B B L 9 Find the volume of the solid obtained by rotating the region bounded by the curve y 2 B and the line y x about the x axis A 1 B 15 1 G 1 H 203 1 I 1 J 23 1 K 1 L 1 M 35 1 N 1 solution The points of intersection are 0 0 and 4 4 Using the disc method 0 B B is further from the line of rotation with distance 2 B and 1 B B is closer with ditance x Then the volume is 1 B B B 1 J 10 Find the volume of the solid obtained by rotating the region bounded by the curve B C and the line B about the line B 21 A 1 B 151 G 1 D 2 1 I 1 J K 1 L 1 M 51 N 1 solution The points of intersection are 1 1 and 1 The distance from 0 C C and the line B is C so using the disc method we get the volume 1 C C 1 C C C 1 C C C 1 M Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curve C B B and the line B C about the C axis A 1 B 1 G 1 H 2 1 I 1 J 2 1 G 1 L 1 M 1 N 1 solution For cylindrical shell about the y axis you must describe the region in terms of functions of x 0 B B B is the top curve and 1 B B is the bottom curve The intersection points are 0 3 and 3 0 For each x between 0 and 3 the distance to the y axis is x Therefore the volume is given by 21 B B B B B 1 B B B 1 F 12 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curve C B and the lines C B about the line C 21 A 1 B 1 1 G 1 H 2 1 I 1 J K 1 L 1 M 1 N 1 solution About the line C we must use the region between function of y 1 1 is the intersection point of B and B C with B the top and B C the bottom For each y between 0 and 1 the distance to C is C Z 1 C C C 1 C C C C 1 F 13 Which of the following integrals gives the length of the curve B C A F G H I J K L M N C C B C F P B solution B 14 Find the average value of y x over the interval 1 3 A F G H I J K L M N solution average value B B H 15 A force of 10 lb is required to hold a spring stretched 5 inches beyond its natural length Find the amount of work done in ft lbs to stretch it from its natural length to 6 inches beyond its natural length A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 solution J B 5B with given J 5 5 J B B B B 0 6 G 16 Suppose 3 ft lbs of work is required to stretch a spring from 2 ft to 4 ft beyond its natural length How far in ft will a weight of 5 lb stretch the spring beyond its natural length A 20 B 18 75 C 15 D 10 E 8 F 6 25 G 5 H 3 5 I 2 J 1 5 solution Given 5B B 5 B l 5 5 J B B Then B gives us that B 0 H PART II CLEARLY WRITE YOUR SOLUTION AND HOW YOU GOT IT 17 Consider the region in the first quadrant bounded by the curve C B and the lines B C and C a Find the area by integrating with respect to x 0 B 1 B B 5 points solution Intersection point in the first quadrant of C B and C B is 2 4 The base of the region is C and the top is C B for 0 x 2 and C B for 0 x 2 The the area needs 2 integrals A B B B B d b Find the area by integrating with respect to y c 0 y 1 y y 5 points Make sure you get the same answer as above solution As functions of y the region has a top of B C and a base of B C 0 C Then the …
View Full Document
Unlocking...